Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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9
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7answers
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Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...
30
votes
2answers
2k views

Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
30
votes
4answers
2k views

Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs. If p=1 mod 4 is a prime, we can define the ...
10
votes
2answers
743 views

Number of isomorphism types of finite groups

Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?
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votes
3answers
1k views

Estimating a partial sum of weighted binomial coefficients

There is a well-known estimate for the sum of all binomial coefficients $\binom{n}{k}$ satisfying $k \leq \alpha n$ for some $\alpha$ satisfying $0 < \alpha \leq 1/2$: $$ \sum_{k=0}^{\alpha ...
57
votes
3answers
4k views

What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...
21
votes
5answers
2k views

Generating finite simple groups with $2$ elements

Here is a very natural question: Q: Is it always possible to generate a finite simple group with only $2$ elements? In all the examples that I can think of the answer is yes. If the answer is ...
34
votes
2answers
2k views

A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
7
votes
2answers
1k views

Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...
9
votes
1answer
355 views

The Simultaneous Conjugacy Problem in the symmetric group $S_N$

We are interested in the following notions in the case $G=S_N$, the symmetric group on $\{1,\dots,N\}$. Fix a group $G$ and a number $d$. For $(g_1,\dots,g_d)\in G^d$ and $x\in G$, define ...
5
votes
1answer
562 views

What is the largest number of k-element subsets of a given n-element set S such that…

Given a set S of n elements. What is the largest number of k-element subsets of S such that every pair of these subsets has at most one common element?
7
votes
1answer
300 views

Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example). There are connected countable graphs that are isomorphic to ...
5
votes
1answer
472 views

Difference Sets

Suppose $$ P \subseteq \{1,2,\dots,N\},\quad |P| = K $$ We calculate the differences as: $$d=p_i-p_j\mod N,\quad i\ne j$$ Now let $a_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N ...
1
vote
1answer
349 views

Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected. The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
59
votes
17answers
48k views

Google question: In a country in which people only want boys [closed]

Hi all! Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is: ...
56
votes
9answers
7k views

Why do Bernoulli numbers arise everywhere?

I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...
27
votes
16answers
7k views

Linear Algebra Proofs in Combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
59
votes
6answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
43
votes
6answers
3k views

How to recognise that the polynomial method might work

A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson. Theorem. Let ...
33
votes
4answers
3k views

Integer-valued factorial ratios

This historical question recalls Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation Chebyshev used the factorial ratio sequence $$ ...
41
votes
2answers
5k views

Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is Möbius nearly Orthogonal to Morse ! Harold Calvin Marston Morse (24 March ...
31
votes
5answers
3k views

Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
23
votes
18answers
8k views

Interesting and Accessible Topics in Graph Theory

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...
30
votes
20answers
4k views

Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...
35
votes
6answers
2k views

Placing numbers $1,2,\ldots,n^3$ in a cube so that numbers of any two adjacent unit subcube are coprime

This is a question first I asked in SE but since there was no suggestion or solution, I decide to put it here. Consider an $n\times n \times n$ Cube containing $n^3$ unit cubes. Is it possible to ...
28
votes
9answers
20k views

Sum of 'the first k' binomial coefficients for fixed n

I am interested in the function $\sum_{i=0}^{k} {N \choose i}$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other notable ...
20
votes
4answers
2k views

Number of vectors so that no two subset sums are equal

Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 ...
25
votes
2answers
1k views

Is there a finite family of functions such that the max of any two functions can be dominated by a third?

Is it true that for every $t$ there is an $n$ and there exists a finite function family, $\cal F$, whose members are from $[n] \to \mathbb N$ (taking all different values) and for any $f_1, \ldots, ...
19
votes
5answers
3k views

Number of valid topologies on a finite set of n elements

I've heard that the problem of counting topologies is hard, but I couldn't really find anything about it on the rest of the internet. Has this problem been solved? If not, is there some feature that ...
15
votes
10answers
3k views

Applications of infinite Ramsey's Theorem (on N)?

Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) ...
18
votes
0answers
396 views

Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...
21
votes
1answer
1k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order ...
14
votes
5answers
1k views

A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all, So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is: for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$ Is there a general ...
41
votes
4answers
4k views

Connectivity of the Erdős–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
32
votes
2answers
719 views

How close can one get to the missing finite projective planes?

This question can be interpreted as an instance of the Zarankiewicz problem. Suppose we have an $n\times n$ matrix with entries in $\{0,1\}$ with no $\begin{pmatrix}1 & 1\\ 1& 1\end{pmatrix} $ ...
19
votes
2answers
902 views

Calculating Mayer-Vietoris efficiently

This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book ...
10
votes
5answers
3k views

infinite permutations

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same ...
7
votes
1answer
1k views

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
21
votes
3answers
2k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot ...
18
votes
1answer
1k views

A geometric series equalling a power of an integer

The following problem cropped up whilst considering generalised quadrangles with a product structure, and it boils down to a simple number theoretic problem. Let $s$ be an integer greater than 2 and ...
17
votes
4answers
1k views

What is the minimum N for which there exist N points in the plane that cannot be covered by any number of non-overlapping closed unit discs?

This problem was posed in March 2010 at G4G9 in a talk by the Japanese mathematician Hirokazu "Iwahiro" Iwasawa. He claims there is a simple proof that N > 10, though he did not share it with the ...
15
votes
2answers
811 views

Minimal graphs with a prescribed number of spanning trees

As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...
10
votes
3answers
955 views

A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form : $$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that : $d_{i}\vert n$ $1=d_{1}<d_{2} \le d_{3} \le ... \le ...
9
votes
2answers
788 views

Is the Steiner ratio Gilbert–Pollak conjecture still open?

Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest network interconnecting $P$ must be a tree, which is called a Steiner minimum ...
16
votes
3answers
1k views

Does subgroup structure of a finite group characterize isomorphism type?

Question Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ ...
7
votes
4answers
2k views

Mean minimum distance for N random points on a unit square (plane)

A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...
6
votes
2answers
2k views

Degree Sequences and Graph Enumeration

I do recreational math from time to time, and I was wondering about a couple of graph enumeration issues. First, is it possible to enumerate all simple graphs with a given degree sequence? Second, ...
7
votes
2answers
758 views

Highbrow interpretations of Stirling number reciprocity

The number ${n \choose k}$ of $k$-element subsets of an $n$-element set and the number $\left( {n \choose k} \right)$ of $k$-element multisets of an $n$-element set satisfy the reciprocity formula ...
3
votes
1answer
1k views

Counting graphs

I'd love your help with this question. Let $n\geq3$ be a fixed integer. How many non-isomorphic graphs with $p$ vertices and $q$ edges are there where $p+q=n$? Thank you very much.
11
votes
0answers
413 views

Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers

Definition / Question Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition ...